# Polynomials in One Variable

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Let us begin by recalling that a variable is denoted by a symbol that can take any real value. We use the letters x, y, z, etc. to denote variables. Notice that 2x, 3x, – x, -

However, there is a difference between a letter denoting a constant and a letter denoting a variable. The values of the constants remain the same throughout a particular situation, that is, the values of the constants do not change in a given problem, but the value of a variable can keep changing.

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Now, consider a square of side 3 units. What is its perimeter? You know that the perimeter of a square is the sum of the lengths of its

Here, each side is 3 units. So, its perimeter is 4 × 3, i.e.,

What will be the perimeter if each side of the square is 10 units?

The perimeter is 4 × 10, i.e.,

In case the length of each side is x units (see Fig. 2.2), the perimeter is given by 4x units. So, as the length of the side varies, the perimeter varies.

Can you find the area of the square PQRS? It is x × x =

**polynomials in one variable**.

In the examples above, the variable is x. For instance,

In the polynomial **terms** of the polynomial.

Similarly, the polynomial

Each term of a polynomial has a **coefficient.** So, in

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Let us consider these examples

In fact, -2, 7 are examples of constant polynomials. The constant polynomial 0 is called the **zero polynomial.**

**Example : **. This plays a very important role in the collection of all polynomials, as you will see in the higher classes.

**Drop the zero and nonzero polynomials into the concerned boxes.**

Now, consider algebraic expressions such as x +

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A polynomial can have any (finite) number of terms. For instance,

Consider the polynomials 2x, 2,**monomials** (‘mono’ means ‘one’).

Now observe each of the following polynomials:p(x) = x + 1,

How many terms are there in each of these? Each of these polynomials has only **binomials** (‘bi’ means ‘two’).

Similarly, polynomials having only **trinomials** (‘tri’ means ‘three’). Some examples of trinomials are

p(x) = x +

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**Drop the monomials,binomials and trinomials into the concerned boxes.**

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Now, look at the polynomial p(x) = 3

What is the term with the highest power of x ? It is

Similarly, in the polynomial q(y)= 5

We call the highest power of the variable in a polynomial as the **degree of the polynomial**. So, the degree of the polynomial 3**The degree of a non-zero constant polynomial is zero.**

## Example

**Find the degree of each of the polynomials given below**:

**(i)**x 5 -x 4 + 3

**(ii) 2-**y 2 - y 3 + 2 y 8

**(iii) 2**

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Now observe the polynomials p(x) = 4x + 5, q(y) = 2y, r(t) = t + and s(u) = 3 – u. Do you see anything common among all of them? The degree of each of these polynomials is one. A polynomial of degree one is called a

Some more linear polynomials in one variable are 2x – 1, y + 1, 2 – u. Now, try and find a linear polynomial in x with 3 terms? You would not be able to find it because a linear polynomial in x can have at most two terms. So, any linear polynomial in x will be of the form ax + b, where a and b are constants and a ≠ 0 (why?). Similarly, ay + b is a linear polynomial in y.

Now consider the polynomials :

Do you agree that they are all of degree two? A polynomial of degree two is called a

If you observe any quadratic polynomial in x then it is of the form

We call a polynomial of degree

How many terms do you think a cubic polynomial in one variable can have?

It can have at most

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Now, that you have seen what a polynomial of degree 1, degree 2, or degree 3 looks like, can you write down a polynomial in one variable of degree n for any natural number n? A polynomial in one variable x of degree n is an expression of the form.

In particular, if **zero polynomial,** which is denoted by 0. What is the degree of the zero polynomial? The degree of the zero polynomial is

So far we have dealt with polynomials in one variable only. We can also have polynomials in more than one variable. For example,